4.3.1 A One-Operator Algorithm (Reproduction)
In this subsection, the nature of the state transition matrix is examined for the case
of no crossover or mutation (i.e. Q] = (0,0) for 0 < k < f). In this case, the conditional
probability of selecting a solution i E S from a population described by the state vector
ne S' is (i.e. proportional reproduction)
n(i) x R(i)
Vie S,Vn E S' :P,(i[ in)= Eq. 4.2
Y no) x R(j)
j S
where the subscript 1 indicates that the one-operator case is under consideration. Thus,
the conditional probability of the successor generation described by m given that the pres-
ent generation is described by n is a multinomial distribution, i.e.
M!
Vm,n e S': P,(m I n) = x P,(i I n)ri)
H m(i)! ieS
is S
= x P,(i In)m(i) Eq. 4.3
m ie S
M n(i) x R(i) m(i)
(0 Eq. 4.5
jeS
The transition probability matrix of the Markov chain representing the one-operator algo-
rithm is composed of the array of conditional probabilities defined by Eq. 4.3, i.e.
= [P,(m I n)]. Eq. 4.6